Python: create popup dialog from panel?
This turned out to be easier than I thought. You simply do something like this:

import bpy
from bl_ui.properties_paint_common import UnifiedPaintPanel

class BasicSettings(bpy.types.Operator, UnifiedPaintPanel):
    """
    The basic brush settings like strength, radius etc.
    """
    bl_idname = "view3d.basic_settings"
    bl_label = "Brush Basic Settings"

    def execute(self, context):
        """
        Show the basic brush settings popup dialog.
        """
        return context.window_manager.invoke_popup(self, width=300)

    def draw(self, context):
        """
        What follows is basically a copy of the Python code
        responsible for the placement of the settings in the Tools tab,
        taken from Blender itself.
        """
        layout = self.layout
        toolsettings = context.tool_settings
        settings = self.paint_settings(context)
        brush = settings.brush

        capabilities = brush.sculpt_capabilities

        col = layout.column()

        col.separator()

        row = col.row(align=True)

        ups = toolsettings.unified_paint_settings
        if ((ups.use_unified_size and ups.use_locked_size) or
                ((not ups.use_unified_size) and brush.use_locked_size)):
            self.prop_unified_size(
                row, context, brush,
                "use_locked_size", icon='LOCKED')
            self.prop_unified_size(
                row, context, brush, "unprojected_radius", slider=True, text="Radius")
        else:
            self.prop_unified_size(
                row, context, brush,
                "use_locked_size", icon='UNLOCKED')
            self.prop_unified_size(
                row, context, brush,
                "size", slider=True, text="Radius")


Etc. These are just copy pasted from within Blender. Just go to any UI element in the panel you want, right click on click and then Edit Source, it will open a file in the text editor and from that you have to figure out what you want and then just copy paste it. And now, whenever you call this operator shown here, it will create a popup dialog similar to the panel.